Heat grill to high. Divide the ground turkey into 4 equal burgers, brush with oil, and season with salt and pepper on both sides. Grill for 3 to 4 minutes per side for medium-well doneness (the burger will continue to cook when placed back on the grill in the next step).
Place the bread on a flat surface. Top each slice with a slice of cheese and a few tablespoons of Grilled Onion Relish. Place a burger on top and then top the burger with the remaining 4 slices of bread. Brush the top of the bread with the oil and place the burgers carefully on the grill, oiled side down, and press down slightly on the sandwich with a metal spatula. Grill for 1 to 2 minutes or until golden brown, then brush the top slice of bread with oil, flip over, press the top with the spatula and cook until that side is golden brown and the cheese has melted. Cut in half and serve.
Heat grill to high. Brush onions on both sides with oil and season with salt and pepper. Grill on each side for 3 to 4 minutes or until slightly charred and just cooked through.
Whisk together vinegar, mustard, salt, pepper, and thyme in a medium bowl. Whisk in 1/4 cup of olive oil until emulsified. Add the onions and stir until combined. Let stand at room temperature until ready to use.